– Gradients, derivatives, second derivatives, derivative from first principle
\(\frac{dy}{dx}\) is the rate of change of y with respect to x
\(\frac{dy}{dx}\) = f'(x) and \(\frac{d^2y}{dx^2}\) = f”(x)
Differentiation from first principle is
\(\frac{dy}{dx}\) = lim \(\frac{f(x +h)-f(x)}{h}\)
Sometimes, h=∆x or δx
– Derivative of \(x^n\)
\(\frac{dy}{dx}\) = lim \(\frac{(x +h)^n-(x)^n}{h}\) = nxn-1
– Gradients, Tangents and Normals ; Maxima and Minima
For a local maximum or a local minimum f'(x)=0
f ” (x)> 0 implies a minimum
f ”(x)<0 implies a maximum
f ”(x)= 0 is a point of inflection
– Product rule, Quotient Rule, Chain rule
Product rule is (u(x)v(x))’ = u'(x)v(x) -u(x)v'(x)
Quotient rule is \(\frac{u(x)}{v(x)}\) ‘ =
– Functions defined parametrically
The cycloid is defined by the parametric equations
The slope of the tangent at any point is \(\frac{dy}{dx}\) = \(\frac{sinθ}{1- cosθ}\)
– Differential equations and applications
\(\frac{dy}{dx}\) = 3x +2 The solution is y = \(\frac{3}{2}x^2\) + 2x +C
\(\frac{dy}{dx}\) = \(5x^3y^3\) , The solution is y = ± \(\sqrt{\frac{3}{2(5x^3 + C)}}\)